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\title{Weak Nuclear Statistical Equilibrium and the Production of Neutron-Rich Iron-Group Isotopes}

\ShortTitle{Weak Nuclear Statistical Equilibrium}

\author{Tianhong Yu\\
        Clemson Univerisity\\
        E-mail: \email{tyu@clemson.edu}}

\author{\speaker{Bradley S. Meyer}\\
        Clemson Univerisity\\
        E-mail: \email{mbradle@clemson.edu}}

\abstract{Calcium-48, $^{50}$Ti, and $^{54}$Cr are neutron-rich iron-group
isotopes that show roughly correlated excesses and deficits in certain 
calcium-aluminum-rich inclusions (CAIs) in primitive meteorites 
\cite{2006mess.book...69M}.
These isotopes are produced in high-temperature, low-entropy-per-nucleon
environments such that the nuclear population are governed by a 
quasi-statistical equilibrium with too many heavy nuclei compared to nuclear
statistical equilibrium \cite{1996ApJ...462..825M}.
Such environments are present in dense
thermonuclear (Type Ia) supernovae \cite{1997ApJ...476..801W}.
Production of these isotopes also
requires an electron fraction $\rm{Y_e}$ approximately equal to 0.42, which is
set by electron captures during the explosion. We use NucNet Tools, an
open-source suite of tools for nucleosynthesis \cite{nnt}, to study 
nucleosynthesis in high-density, low-entropy environments appropriate for 
Type Ia supernovae and follow the neutronization of the matter by weak
interactions. We study how the nuclear populations evolve towards and into 
dynamical weak statistical equilibrium \cite{2010A&A...522A..25A}.
Interestingly, for temperature
in the range of $\rm{T_9}$ = 4-6, the evolution to weak statistical equilibrium
is via a quasi-equilibrium, not a nuclear statistical equilibrium, because
the timescale to change the number of heavy nuclei is comparable to the 
timescale to change $\rm{Y_e}$. We confirm that the environments that are able 
to generate low enough $\rm{Y_e}$ to produce $^{48}$Ca are extreme and thus 
probably rare in Galactic history. Nevertheless, when they do occur, they 
produce copious amounts of $^{48}$Ca and the other neutron-rich species. For 
this reason, it is likely that the abundance of these isotopes in interstellar
dust is quite heterogeneous. Because the CAIs formed from interstellar
dust precursors, they inherited this heterogeneity.}

\FullConference{XII International Symposium on Nuclei in the Cosmos,\\
		August 5-12, 2012\\
		Cairns, Australia}


\begin{document}

\section{Introduction}

\section{Weak Nuclear Statistical Equilibrium}

For a typical weak nuclear reaction $\beta$-decay 
$i \to j + e^- + \bar{\nu_e}$ we may get the
forward reaction rate $\lambda_{forward}$ from laboratory or 
theoretical nuclear physics models.  The reverse
rates $\lambda_{reverse}$ (e.g. electron capture rate $e^- + j \to i + \nu_e$) 
are generally endothermic in the laboratory
and have to be calculated from nuclear models as well
(e.g., \cite{1985ApJ...293....1F}).  While the models are in general quite
good, there is no guarantee that the forward and reverse rates are related
in such a way that a nuclear reaction network will evolve, given sufficient
time, into weak statistical equilibrium (WSE). An equilibrium means that
after a sufficient time of evolution, the abundances of all species in the 
system present get their stable values which depend only on temperature,
density, electron abundance and nuclear properties of each species (mass,
multiplicity).

In this work,
we are computing reverse weak rates on nuclei from detailed balance, as people
do with the strong and electromagnetic rates.  This permits us to evolve
a reaction network dynamically into the full weak equilibrium.
To demonstrate how to apply detailed balance,
let's consider the $\beta$-decay above. At some certain temerature and density
if the two species $i$ and $j$ are
in weak statistical equilibrium, we should have
\begin{equation} \label{eq:wse_balance}
\lambda_{forward} Y_i^{WSE} = \lambda_{reverse} Y_i^{WSE}
\end{equation}
%
In the equation the superscript WSE stands for the value in weak statistical
equilibrium, and $Y_i$ is defined as number of species $i$ per nucleon:
$
Y_i \equiv \frac{N_i}{N_{nucleon}}
$
which is refered to "abundance" and is a useful quantity in such studies.
So eq(\ref{eq:wse_balance}) can be interpreted as the consumption rate of 
species $i$ equals the production rate at equilibrium. And if we can calculate
both $Y^{WSE}$, the reverse rate then can be computed from the forward one.

To get $Y_i$ from other quantities, we start from the chemical potential of
classical particles (nuclei can be considered as classical particles as the
temperature won't get up to nuclei mass level, $T_9\sim 1,kT\sim 0.1$MeV):
\begin{equation} \label{eq:chem}
%
\mu_i = m_i c^2 + kT \ln \left[ \frac{Y_i\rho N_A}{G_i} 
  \left(\frac{2\pi \hbar^2}{m_i kT}\right)^{3/2} \right]
\end{equation}
%
For WSE condition we have $Y_i^{WSE}$ from eq(\ref{eq:chem}): 
%
\begin{equation} \label{eq:yi_wse}
Y_i^{WSE} = \frac{G_i}{\rho N_A} \left(\frac{m_i kT}{2\pi \hbar^2}\right)^{3/2}
  \rm{e}^{ (\mu_i^{WSE} - m_i c^2)/kT }
\end{equation}
%
From eq(\ref{eq:wse_balance}) and eq(\ref{eq:yi_wse}) we can get the ratio of
reverse and forward rates:
%
\begin{equation} \label{eq:ratio_old}
\frac{\lambda_{reverse}}{\lambda_{forward}} = \frac{Y_i^{WSE}}{Y_j^{WSE}}
  = \frac{G_i}{G_j} \left( \frac{m_i}{m_j} \right)^{3/2}
    \rm{e}^{ (\mu_i^{WSE} - \mu_j^{WSE})/kT } \ 
    \rm{e}^{ -(m_i - m_j)/kT }
\end{equation}
%
The chemical potential for each species is (ref) 
%
\begin{equation} \label{eq:mui}
\mu_i^{WSE} = A_i \mu_n^{WSE} + Z_i \mu_e^{WSE}
\end{equation}
%
and $A_i = A_j$ in this case. Thus,
%
\begin{equation} \label{eq:ratio}
\frac{\lambda_{reverse}}{\lambda_{forward}} =  
  \frac{G_i}{G_j} \left( \frac{m_i}{m_j} \right)^{3/2}
  \rm{e}^{ (Z_j - Z_i) \mu_e^{WSE}/kT } \ 
  \rm{e}^{ -Q/kT }
\end{equation}
%
where $Q = m_i c^2 - m_j c^2$ is just the Q-value (mass difference)
of the reaction. With
libstatmech we can easily compute the chemical potential of electron at 
weak statistical equilibrium and thereby compute reverse rates from forward
ones.

\section{Network Calculation}

Fig. \ref{fig:ye_t} shows the difference between using Langanke rates only and 
using Langanke rates and supplemented with approximate rates, with zero 
neutrino chemical potential.

\begin{figure}
\begin{center}
  \includegraphics[height=.3\textheight]{ps_figures/ye_t.ps}
  \caption{Compare Ye using Langanke rates only and Langanke rates with 
           approximate rates. The neutrino chemical potential is set to be -inf
           in these calculations.}
\label{fig:ye_t}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
  \includegraphics[height=.3\textheight]{ps_figures/expansion_ye_t9.ps}
  \caption{Ye evolves with $T_9$ in an expansion calculation.}
\label{fig:expansion_ye_t9}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
  \includegraphics[height=.3\textheight]{ps_figures/expansion_mass.ps}
  \caption{Mass fractions evolve with $T_9$ in an expansion calculation.}
\label{fig:expansion_mass}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
  \includegraphics[height=.3\textheight]{ps_figures/yedot_t.ps}
  \caption{Compare Ye change rates with time for $\beta^-$ decay and electron 
           capture.}
\label{fig:yedot_t}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
  \includegraphics[height=.3\textheight]{ps_figures/yedot_ye.ps}
  \caption{Compare $Y_e$ change rates with $Y_e$ for $\beta^-$ decay and 
           electron capture.}
\label{fig:yedot_ye}
\end{center}
\end{figure}

\section{Conclusion}

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\bibitem{1997ApJ...476..801W} 
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\bibitem{nnt}
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\bibitem{2010A&A...522A..25A} 
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\bibitem{1985ApJ...293....1F} 
Fuller, G.~M., Fowler, W.~A., \& Newman, M.~J.\ 1985, 
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\end{thebibliography}

\end{document}


